Polynomial-time solvable #CSP problems via algebraic models and Pfaffian circuits

Margulies, Susan S.; Morton, Jason

doi:10.1016/j.jsc.2015.06.008

Cited by 5 publications

(5 citation statements)

References 34 publications

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“…However, this is not the case. As previously mentioned, it was shown in [13] that the SWAP gate is not in the GLp2, Cq 4 orbit of any Pfaffian (co)gate. Thus we need only consider the boundary of these orbits.…”

Section: 2mentioning

confidence: 85%

“…However, it has been shown that this is not the case [13]. Under our extended definition, if there is a circuit Γ in the orbit closure of a Pfaffian circuit which contains the SWAP gate, this implies that the SWAP gate lies in the orbit closure of SLp2, Cq 4 acting on P 4 or P _ 4 .…”

Section: Orbit Closures Of Pfaffian (Co)gatesmentioning

confidence: 91%

“…The map that allows two wires to be switched is called the SWAP gate (or tensor) and acts on a basis of C 2 b C 2 by e i b e j Þ Ñ e j b e i . It is known that this tensor is not a Pfaffian gate in any basis [13]. However, it may the case that a combination Pfaffian gates in some basis can be composed to form the SWAP gate; this question is still open.…”

Section: Introductionmentioning

confidence: 99%

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Tensors masquerading as matchgates: Relaxing planarity restrictions on Pfaffian circuits

Turner

2017

Journal of Computer and System Sciences

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TURNERÅbstract. Holographic algorithms, alternatively known as Pfaffian circuits, have received a great deal of attention for giving polynomial-time algorithms of #P-hard problems. Much work has been done to determine the extent of what this machinery can do and the expressiveness of these circuits. One aspect of interest is the fact that these circuits must be planar. Work has been done to try and relax the planarity conditions and extend these algorithms further. We show that an approach based on orbit closures does not work, but give a different technique for allowing the SWAP gate to be used in a Pfaffian circuit given a suitable basis and restricted type of graph. This is done by exploiting the fact that the set of Pfaffian (co)gates always lies in a hyperplane. We then give a variety of bases that can be chosen such that the SWAP gate acts like a Pfaffian cogate and discuss how many SWAP gates can be implemented in a Pfaffian circuit.

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Section: 2mentioning

confidence: 85%

Section: Orbit Closures Of Pfaffian (Co)gatesmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Tensors masquerading as matchgates: Relaxing planarity restrictions on Pfaffian circuits

Turner

2017

Journal of Computer and System Sciences

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“…The possibility that accidental or freak objects in the enumeration exist cannot be discounted if the objects in the enumeration have not been studied systematically." Indeed, if any "freak" object exists in this framework, it would collapse #P to P. Therefore, over the past 10 to 15 years, this technique has been intensely studied in order to gain a systematic understanding of the limit of the trio of holographic reductions, matchgates, and the FKT algorithm [6,7,15,16,32,37,38,42,46]. Without settling the P versus #P question, the best hope is to achieve a complexity classification.…”

Section: Introductionmentioning

confidence: 99%

FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints

Cai

Guo

et al. 2021

Theory Comput Syst

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We prove a complexity classification for Holant problems defined by an arbitrary set of complex-valued symmetric constraint functions on Boolean variables. This is to specifically answer the question: Is the Fisher-Kasteleyn-Temperley (FKT) algorithm under a holographic transformation (Valiant, SIAM J. Comput. 37(5), 1565–1594 2008) a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable on general graphs? There are problems that are #P-hard on general graphs but polynomial-time solvable on planar graphs. For spin systems (Kowalczyk 2010) and counting constraint satisfaction problems (#CSP) (Guo and Williams, J. Comput. Syst. Sci. 107, 1–27 2020), a recurring theme has emerged that a holographic reduction to FKT precisely captures these problems. Surprisingly, for Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false. A dichotomy theorem for #CSPd, which denotes #CSP where every variable appears a multiple of d times, has been an important tool in previous work. However the proof for the #CSPd dichotomy violates planarity, and it does not generalize to the planar case easily. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSPd dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP2. In this case, the putative form of the dichotomy is true. (This is presented in Part II of the paper.) We manage to prove the planar Holant dichotomy relying only on this planar #CSP2 dichotomy, without resorting to a more general planar #CSPd dichotomy for d ≥ 3. A special case of the new polynomial-time computable problems is counting perfect matchings (#PM) over k-uniform hypergraphs when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

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“…This includes the question of both classifying and recognizing Holant problems, holographic circuits, and matchgates (cf. [29,37]). Recognizing a holographic circuit in an arbitrary basis is still an open problem as they are formulated with respect to a specific basis.…”

Section: Introductionmentioning

confidence: 99%

A Finite-Tame-Wild Trichotomy Theorem for Tensor Diagrams

Turner¹

2017

Preprint

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In this paper, we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis. In particular, to a string diagram in a certain monoidal category (which we call tensor diagrams), we formulate an associated abelian category of representations. Each representation corresponds to a tensor network on that diagram. We then classify which tensor diagrams give rise to categories that are finite, tame, or wild in the traditional sense of representation theory. For those tensor diagrams of finite and tame type, we classify the indecomposable representations. Our main result is that a tensor diagram is wild if and only if it contains a vertex of degree at least three. Otherwise, it is of tame or finite type.

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Polynomial-time solvable #CSP problems via algebraic models and Pfaffian circuits

Cited by 5 publications

References 34 publications

Tensors masquerading as matchgates: Relaxing planarity restrictions on Pfaffian circuits

Tensors masquerading as matchgates: Relaxing planarity restrictions on Pfaffian circuits

FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints

A Finite-Tame-Wild Trichotomy Theorem for Tensor Diagrams

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